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Ukrainian Mathematical Journal

, Volume 66, Issue 5, pp 645–665

# On the Behavior of Algebraic Polynomial in Unbounded Regions with Piecewise Dini-Smooth Boundary

• F. G. Abdullayev
• P. Özkartepe
Article
Let G ⊂  be a finite region bounded by a Jordan curve L := ∂G, let $$\Omega :=\mathrm{e}\mathrm{x}\mathrm{t}\overline{G}$$ (with respect to $$\overline{\mathbb{C}}$$), let Δ := {w : |w| > 1}, and let w = Φ(z) be the univalent conformal mapping of Ω onto Δ normalized by Φ (∞) = ∞, Φ′(∞) > 0. Also let h(z) be a weight function and let A p (h,G), p > 0 denote a class of functions f analytic in G and satisfying the condition
$${\left\Vert f\right\Vert}_{A_p\left(h,G\right)}^p:={\displaystyle \int {\displaystyle \underset{G}{\int }h(z){\left|f(z)\right|}^pd{\sigma}_z<\infty, }}$$
where σ is a two-dimensional Lebesgue measure.
Moreover, let P n (z) be an arbitrary algebraic polynomial of degree at most n ∈ ℕ. The well-known Bernstein–Walsh lemma states that
$$\begin{array}{cc}\hfill \left|{P}_n(z)\right|\le {\left|\varPhi (z)\right|}^n{\left\Vert {P}_n\right\Vert}_{C\left(\overline{G}\right)},\hfill & \hfill z\in \Omega .\hfill \end{array}$$
(*)

In this present work we continue the investigation of estimation (*) in which the norm $${\left\Vert {P}_n\right\Vert}_{C\left(\overline{G}\right)}$$ is replaced by $${\left\Vert {P}_n\right\Vert}_{A_p\left(h,G\right)},p>0$$, for Jacobi-type weight function in regions with piecewise Dini-smooth boundary.

## Keywords

Orthogonal Polynomial Quasiconformal Mapping Jordan Curve Algebraic Polynomial Unbounded Region
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

1. 1.
F. G. Abdullayev and V. V. Andrievskii, “On the orthogonal polynomials in domains with K-quasiconformal boundary,” Izv. Akad. Nauk Azerb. SSR, Ser. FTM, 1, 3–7 (1983).Google Scholar
2. 2.
F. G. Abdullayev, Dissertation (Ph. D.), Donetsk (1986), 120 p.Google Scholar
3. 3.
F. G. Abdullayev, “On the some properties of the orthogonal polynomials over a region in the complex plane (Part III),” Ukr. Math. J., 53, No. 12, 1934–1948 (2001).
4. 4.
F. G. Abdullayev and D. Uğur, “On the orthogonal polynomials with weight having singularity on the boundaries of regions in the complex plane,” Bull. Belg. Math. Soc., 16, No. 2, 235–250 (2009).
5. 5.
L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, Princeton, NJ (1966).
6. 6.
V. V. Andrievskii, V. I. Belyi, and V. K. Dzyadyk, Conformal Invariants in Constructive Theory of Functions of Complex Plane, World Federation Publ. Co., Atlanta (1995).Google Scholar
7. 7.
V. V. Andrievskii and H. P. Blatt, Discrepancy of Signed Measures and Polynomial Approximation, Springer, New York, Inc. (2010).Google Scholar
8. 8.
G. M. Goluzin, Geometric Theory of Functions of Complex Variable [in Russian], Gostekteorhizdat, Moscow–Lenigrad (1952).Google Scholar
9. 9.
E. Hille, G. Szegö, and J. D. Tamarkin, “On some generalization of a theorem of A. Markoff,” Duke Math. J., 3, 729–739 (1937).
10. 10.
O. Lehto and K. I. Virtanen, Quasiconformal Mapping in the Plane, Springer, Berlin (1973).
11. 11.
Ch. Pommerenke, Boundary Behavior of Conformal Maps, Springer, Berlin (1992).
12. 12.
S. Rickman, “Characterisation of quasiconformal arcs,” Ann. Acad. Sci. Fenn. Ser. A. Math., 395, 30 p. (1966).Google Scholar
13. 13.
N. Stylianopoulos, “Strong asymptotics for Bergman polynomials over domains with corners and applications,” Const. Approxim., 38, 59–100 (2013).
14. 14.
J. L. Walsh, “Interpolation and approximation by rational functions in the complex domain,” Amer. Math. Soc. (1960).Google Scholar

## Copyright information

© Springer Science+Business Media New York 2014

## Authors and Affiliations

• F. G. Abdullayev
• 1
• P. Özkartepe
• 1
1. 1.Mersin UniversityMersinTurkey